This is a practical and very soft question, with the combinatorial database http://www.findstat.org in mind.
I have a few, around 20, families of combinatorial objects, for example Dyck paths, permutations, perfect matchings, graphs, etc., together with a few, around 200, maps between them. The maps need not be injective or surjective or have any special properties. (Except perhaps that they appear in the literature and are therefore "interesting".)
Some examples of such maps might be the reversal of a Dyck path, various classical maps that send a Dyck path to a 321-avoiding permutation, the map that sends a permutation to its shape under the Robinson-Schensted correspondence, etc.
Thus, we have (a very small) category, which I'd like to visualize.
The aspect that makes this interesting and non-trivial is, that these maps satisfies numerous identities: many maps are involutions, some are idempotent, many maps commute with other maps or are conjugate to other maps, etc. I have all these data.
So, more precisely: I'm looking for a way to visualize (graphically!) these identities.